\chapter{Fourier-Mukai transforms for DQ-modules}
\selectlanguage{english}
\section{Introduction}

Fourier-Mukai transforms have been extensively studied in algebraic geometry. A key results of Orlov (see \cite{Orlov}) states that any equivalence of triangulated categories between the bounded derived categories of coherent sheaves on smooth projective varieties is  isomorphic to a Fourier-Mukai functor. We refer the reader to the seminal book of Huybrechts \cite{Huy} for an in depth treatment of Fourier-Mukai transforms in algebraic geometry.
In \cite{KS3}, Kashiwara and Schapira have developed a framework to study integral transforms in the setting of DQ-modules. Recently, Arinkin, Block and Pantev have proved in \cite{Mukai_Pantev} that under some condition an integral transform between complex manifolds can be lifted to the non-commutative level.

The aim of this short chapter is to prove Theorem \ref{finalmukai} which says that for any two smooth complex projective varieties $X_1, \; X_2$ equipped respectively with DQ-algebroid stacks $\cA_1, \cA_2$ and any kernel $\cK \in \Der_{\coh}^b(\cA_{12^a})$, the integral transforms $\Phi_\cK: \Der^b_{\coh}(\cA_2) \to \Der^b_{\coh}(\cA_1)$ associated to $\cK$ is is fully faithful (resp. an equivalence of triangulated categories) if and only if $\Phi_{\gr_\hbar \cK}:\Der^b_{\coh}(\cO_2) \to \Der^b_{\coh}(\cO_1)$ is fully faithful (resp. an equivalence of triangulated categories).

We borrow the notation of Chapter 3, especially Notation \ref{not:www1}.

\section{Duality for DQ-modules}
Let $X_i$ $(i=1, \, 2, \, 3)$ be smooth projective complex varieties endowed with the Zariski topology and let $\cA_i$ be a DQ-algebroid on $X_i$. 
We recall some duality results for DQ-modules from \cite[Chap. 3]{KS3}. 

First we need the following result.
\begin{prop} [{\cite[p. 93]{KS3}}]
Let $\cK_i \in \Der^b(\cA_{i(i+1)^a})$ $(i=1 \, ,2)$ and let $\mathcal{L}$ be a bi-invertible $\cA_2 \otimes \cA_{2^a}$-module. Then, there is a natural isomorphism
\begin{equation*}
(\cK_1 \underset{2}{\circ} \mathcal{L})\underset{2}{\circ} \cK_2 \simeq \cK_1 \underset{2}{\circ} (\mathcal{L}\underset{2}{\circ} \cK_2).
\end{equation*}
\end{prop}

%\begin{lemme}
%For $\cK_i \in \Der^b(\cA_{i(i+1)^a})$ $(i=1 \, ,2)$, we have a natural morphism in $\Der^b(\cA_{13})$:
%\begin{equation*}
%(\Du^{\prime}_{\cA_{12}}\cK_1) \underset{2^a}{\circ} \w_{2^a} \underset{2^a}{\circ} (\Du^{\prime}_{\cA_{23}}\cK_2) \to \Du^{\prime}_{\cA_{13}}(\cK_1 \underset{2}{\circ} \cK_2)
%\end{equation*}
%\end{lemme}

We denote by $\w_i$ the dualizing complexe for $\cA_i$. It is a bi-invertible $(\cA_i \otimes \cA_{i^a})$-module. Since the category of bi-invertible $(\cA_i \otimes \cA_{i^a})$-modules is equivalent to the category of coherent $\cA_{ii^a}$-modules simple along the diagonal, we will regard $\w_i$ as an $\cA_{ii^a}$-module supported by the diagonal and we will still denote it by $\w_i$.

\begin{thm}[{\cite[Theorem 3.3.3]{KS3}}] \label{dualdqserre}
Let $\cK_i \in \Der_{\coh}^b(\cA_{i (i+1)^a})$ $(i=1, \, 2)$. %Assume that $\Supp(\cK_1) \times_{X_2} \Supp(\cK_2)$ is proper over $X_1 \times X_{3^a}$. 
There is a natural isomorphism in $\Der^b_{\coh}(\cA_{1^a3})$
\begin{equation*}
(\Du^{\prime}_{\cA_{12^a}}\cK_1) \underset{2^a}{\circ} \w_{2^a} \underset{2^a}{\circ} (\Du^{\prime}_{\cA_{23^a}}\cK_2) \stackrel{\sim}{\to} \Du^{\prime}_{\cA_{13^a}}(\cK_1 \underset{2}{\circ} \cK_2).
\end{equation*}
\end{thm}

\section{Fourier-Mukai functors in the quantized setting} 

Following \cite{KS3}, we define Fourier-Mukai functors in the framework of DQ-modules. For a closed subset $\Lambda$ of $X$ , we denote by $\Der_{\coh, \Lambda}^{b}(\cA_X)$ the full triangulated subscategory of $\Der_{\coh}^{b}(\cA_X)$ consiting of objects supported by $\Lambda$. Recall the following theorem.

\begin{thm}[Theorem 3.2.1 of \cite{KS3}] \label{coherence}
For $i=1 \, ,2$, let $\Lambda_i$ be a closed subset of $X_i \times X_{i+1}$ and $\cK_i \in \Der_{\coh, \Lambda_i}^b(\cA_{i (i+1)^a})$. Assume that $\Lambda_1 \times_{X_2} \Lambda_2$ is proper over $X_1 \times X_3$, and set $\Lambda=p_{13}(p_{12}^{-1} \Lambda_1 \cap p_{23}^{-1} \Lambda_2)$. Then the object $\cK_1 \underset{2}{\circ} \cK_2$ belongs to $\Der^b_{\coh, \Lambda}(\cA_{13^a})$.
\end{thm}  

From now on, all the varieties considered are smooth complex projective varieties endowed with the Zariski topology.
Let $\cK \in \Der^b_{\coh}(\cA_{12^a})$. Theorem \ref{coherence} implies that the functor (\ref{MukaiDQ}) is well-defined.

\begin{align} \label{MukaiDQ}
\Phi_\cK: \Der^b_{\coh}(\cA_2) \to \Der^b_{\coh}(\cA_1)&,& \cM \mapsto \cK \underset{2}{\circ} \cM = Rp_{1\ast}(K \Lte_{p_2^{-1}\cA_2} p_2^{-1} \cM). 
\end{align}

\begin{prop}
Let $\cK_1 \in \Der^b_{\coh}(\cA_{12^a})$ and $\cK_2 \in \Der^b_{\coh}(\cA_{23^a})$. The composition
\begin{align*}
\Der^b_{\coh}(\cA_3)\stackrel{\Phi_{\cK_2}}{\rightarrow}\Der^b_{\coh}(\cA_2) \stackrel{\Phi_{\cK_1}}{\rightarrow}\Der^b_{\coh}(\cA_1)
\end{align*}
is isomorphic to $\Phi_{\cK_1 \underset{2}{\circ} \cK_2}:\Der^b_{\coh}(\cA_3) \to \Der^b_{\coh}(\cA_1)$.
\end{prop}

\begin{proof}
It is a direct consequence of Proposition \ref{assodqmod}.
\end{proof}


\begin{defi}
For any object $\cK \in \Der^b_{\coh}(\cA_{12^a})$, we set
\begin{align*}
\cK_R= \Du^{\prime}_{\cA_{12^a}}(\cK) \underset{2^a}{\circ} \w_{2^a} && \cK_L= \Du^{\prime}_{\cA_{12^a}}(\cK) \underset{1}{\circ} \w_{1}
\end{align*}
objects of $\Der_{\coh}^b(\cA_{1^a2})$.
\end{defi}

\begin{prop}\label{adjoint_mukai}
Let $\Phi_\cK:\Der^b_{\coh}(\cA_2) \to \Der^b_{\coh}(\cA_1)$ be the Fourier-Mukai functor associated to $\cK$ and $\Phi_{\cK_R}:\Der^b_{\coh}(\cA_1) \to \Der^b_{\coh}(\cA_2)$ (resp. $\Phi_{\cK_L}:\Der^b_{\coh}(\cA_1) \to \Der^b_{\coh}(\cA_2)$) the Fourier-Mukai functor associated to $\cK_R$ (resp. $\cK_L$). Then $\Phi_{\cK_R}$ (resp. $\Phi_{\cK_L}$) is right (resp. left) adjoint to $\Phi_\cK$.
\end{prop}

\begin{proof}
We have
\begin{equation*}
\RHom_{\cA_1}(\cK \underset{2}{\circ} \cM, \cN) \simeq \Rg(X_1,\fRHom_{\cA_1}(\cK \underset{2}{\circ} \cM, \cN)).
\end{equation*}

Applying Theorem \ref{dualdqserre} and the projection formula, we get

\begin{align*}
\fRHom_{\cA_1}(\cK \underset{2}{\circ} \cM, \cN) & \simeq \fRHom_{\cA_1}(\cK \underset{2}{\circ} \cM, \cA_1) \Lte_{\cA_1} \cN\\
&\simeq (\Du^{\prime}_{\cA_{12^a}}(\cK)\underset{2^a}{\circ} \w_{2^a} \underset{2^a}{\circ} \Du^{\prime}_{\cA_2}(\cM)) \Lte_{\cA_1} \cN\\
&\simeq (\cK_R \underset{2^a}{\circ} \Du^{\prime}_{\cA_2}(\cM)) \Lte_{\cA_1} \cN\\
&\simeq \dR p_{1 \ast} ( \cK_R \Lte_{p_2^{-1} \cA_{2^a}} p_2^{-1} \Du^{\prime}_{\cA_2}(\cM)) \Lte_{\cA_1} \cN\\
&\simeq \dR p_{1 \ast} ( \cK_R \Lte_{p_2^{-1} \cA_{2^a}} p_2^{-1} \Du^{\prime}_{\cA_2}(\cM) \Lte_{p_1^{-1}\cA_1} p_1^{-1} \cN).
\end{align*}

Taking the global section and applying again the projection formula, we get
\begin{align*}
\Rg(X_1,\fRHom_{\cA_1}(\cK \underset{2}{\circ} \cM, \cN))& \simeq \Rg(X_1, \dR p_{1 \ast} ( \cK_R \Lte_{p_2^{-1} \cA_{2^a}} p_2^{-1} \Du^{\prime}_{\cA_2}(\cM)) \Lte_{p_1^{-1}\cA_1} p_1^{-1} \cN)\\
%
&\simeq \Rg(X_1 \times X_2,(\cK_R \Lte_{p_2^{-1} \cA_{2^a}} p_2^{-1} \Du^{\prime}_{\cA_2}(\cM)) \Lte_{p_1^{-1}\cA_1} p_1^{-1} \cN)\\
%
& \simeq \Rg(X_2, \dR p_{2 \ast} ( (\cK_R \Lte_{p_2^{-1} \cA_{2^a}} p_2^{-1} \Du^{\prime}_{\cA_2}(\cM)) \Lte_{p_1^{-1}\cA_1} p_1^{-1} \cN))\\
%
&\simeq \Rg(X_2,\Du^{\prime}_{\cA_2}(\cM) \Lte_{\cA_2} (\cK_R \underset{1}{\circ} \cN))\\
&\simeq \RHom_{\cA_2}(\cM,\cK_R \underset{1}{\circ} \cN).
\end{align*}
Thus, $\RHom_{\cA_1}(\cK \underset{2}{\circ} \cM, \cN)\simeq \RHom_{\cA_2}(\cM,\cK_R \underset{1}{\circ} \cN)$ which proves the claim. The proof is similar for $\cK_L$.
\end{proof}

%\begin{prop}\label{stepmukaiun}
%Let $X_1$ (resp. $X_2$) be compact complex manifolds endowed with a DQ-algebroid $\cA_1$ (resp. $\cA_2$). Let $\cK \in \Der^b_{\coh}(\cA_{12^a})$. Let $\Phi_\cK: \Der^b_{\coh}(\cA_2) \to \Der^b_{\coh}(\cA_1)$. If the functor $\Phi_{\gr_\hbar \cK}:\Der^b_{\coh}(\cO_2) \to \Der^b_{\coh}(\cO_1)$ is an equivalence of triangulated categories (resp. fully faithful), then the functor $\Phi_\cK$ is an equivalence of triangulated categories (resp. fully faithful). 
%\end{prop}
%
%\begin{proof}
%Since $\Phi_\cK$ and $\Phi_{\cK_R}$ are adjoint functors we have natural morphisms of functors
%\begin{align*}
%\Phi_{\cK} \circ \Phi_{\cK_R} \to \id \\
%\id \to \Phi_{\cK_R} \circ \Phi_{\cK}.
%\end{align*}
%
%If $\cM \in \Der^b_{\coh}(\cA_1)$, then 
%
%\begin{equation} \label{buttraite}
%\Phi_{\cK} \circ \Phi_{\cK_R}(\cM) \to \cM \\
%\end{equation}
%
%Applying the functor $\gr_\hbar$, we get
%
%\begin{equation}\label{equicom}
%\Phi_{\gr_\hbar \cK} \circ \Phi_{\gr_\hbar \cK_R}(\gr_\hbar \cM) \to (\gr_\hbar \cM).
%\end{equation}
%
%If $\Phi_{\gr_\hbar \cK}$ is an equivalence of triangulated category, then the morphism \ref{equicom} is an isomorphism. The objects $\Phi_{\cK} \circ \Phi_{\cK_R}(\cM)$ and $\cM$ are cohomologically complete since they belongs to $\Der^b_{\coh}(\cA_1)$.
%Thus the morphism (\ref{buttraite}) is an isomorphism that is to say
%\begin{equation*}
%\Phi_{\cK} \circ \Phi_{\cK_R} \stackrel{\sim}{\to} \id.
%\end{equation*}
%Similarly, one shows that if $\Phi_{\gr_\hbar \cK}$ is an equivalence then
%\begin{equation*}
%\id \stackrel{\sim}{\to} \Phi_{\cK_R} \circ \Phi_{\cK}.
%\end{equation*}
%It follows that $\Phi_\cK$ is an equivalence of categories.
%
%%\begin{align*}
%%\Phi_{\gr_\hbar \cK} \circ \Phi_{\gr_\hbar \cK_R} \to \Phi_{\cO_{\Delta_1}}\simeq \id \\
%%\Phi_{\gr_\hbar \cK_R} \circ \Phi_{\gr_\hbar \cK} \to \Phi_{\cO_{\Delta_2}} \simeq \id
%%\end{align*}
%\end{proof}

%Recall the following important results due to Bondal and Van den Bergh.
%
%\begin{thm} \label{BVdBcoh}
%Let $X$ be a smooth algebraic variety over a field. 
%\begin{enumerate}[(i)]
%\item The category $\Der_{\qcoh}(\cO_X)$ is compactly generated by a single perfect complex $\mathcal{G}$.
%\item The category $\Der^b_{\coh}(\cO_X)$ is strongly finitely generated by $\mathcal{G}$.
%\end{enumerate}
%\end{thm}
%
%\begin{Rem}
%The Theorem \ref{BVdBcoh} means that $\Der^b_{\coh}(\cO_X)=\langle \mathcal{G} \rangle$, where $\langle \mathcal{G} \rangle$ denotes the smallest full triangulated subcategory of $\Der^b_{\coh}(\cO_X)$ stable under isomorphisms, direct summands and containing $\mathcal{G}$.
%\end{Rem}


%\begin{prop}\label{stepmukaideux}
%Let $X_1$ (resp. $X_2$) be a smooth complex projective variety endowed with a DQ-algebroid $\cA_1$ (resp. $\cA_2$). Let $\cK \in \Der^b_{\coh}(\cA_{12^a})$. Let $\Phi_\cK: \Der^b_{\coh}(\cA_2) \to \Der^b_{\coh}(\cA_1)$. If the functor $\Phi_\cK$ is an equivalence of triangulated categories (resp. fully faithful), then the functor $\Phi_{\gr_\hbar \cK}:\Der_{\coh}(\cO_2) \to \Der^b_{\coh}(\cO_1)$ is an equivalence of triangulated categories (resp. fully faithful). 
%\end{prop}
%
%\begin{proof}
%The functor $\Phi_{\gr_\hbar \cK_R}$ is a right adjoint of the functor $\Phi_{\gr_\hbar \cK}$. Thus, there are morphisms of functors
%\begin{align}
%\Phi_{\gr_\hbar \cK} \circ \Phi_{\gr_\hbar \cK_R} \to \id \label{isoeqi1}\\
%\id \to \Phi_{\gr_\hbar \cK_R} \circ \Phi_{\gr_\hbar \cK} \label{isoeqi2}
%\end{align}
%
%Let $\mathcal{T}_1$ be the full subcategory of $\Der^b_{\coh}(\cO_1)$ whose objects are the $\cM \in \Der^b_{\coh}(\cO_1)$ such that
%\begin{equation*}
%\Phi_{\gr_\hbar \cK} \circ \Phi_{\gr_\hbar \cK_R} (\cM) \to \cM
%\end{equation*}
%is an isomorphism.
%
%The category $\mathcal{T}_1$ is non-empty since $0$ belongs to it and is stable by shifts. Moreover, $\mathcal{T}_1$ is triangulated. Indeed, $\mathcal{T}_1$ is a full subcategory of the triangulated category $\Der_{\coh}^b(\cO_1)$, the only thing to check is that $\mathcal{T}_1$ is stable by taking cones.
%
%Let $f: \cM \to \cN$ a morphism of $\mathcal{T}_1$ and set $\Phi_{\cL}=\Phi_{\gr_\hbar \cK} \circ \Phi_{\gr_\hbar \cK_R}$. Consider a distinguished triangle in $\Der_{\coh}^b(\cO_1)$
%%
%\begin{equation*}
%\cM \stackrel{f}{\to} \cN \to \mathcal{Q} \to \cM[1].
%\end{equation*}
%%
%We have the following diagram
%%
%\begin{equation*}
%\xymatrix{
%\Phi_{\cL}(\cM) \ar[r]^-{\Phi_{\cL}(f)} \ar[d]^-{\wr} & \Phi_{\cL}(\cN) \ar[r] \ar[d]^-{\wr} & \Phi_{\cL}(\mathcal{Q}) \ar[r] \ar[d]& \Phi_{\cL}(\cM[1]) \ar[d]^-{\wr}\\
%%
%\cM \ar[r]^-{f} & \cN \ar[r] & \mathcal{Q} \ar[r] & \cM[1]
%}
%\end{equation*}
%By the five Lemma it follows that the morphism $\Phi_{\cL}(\mathcal{Q}) \to \mathcal{Q}$ is an isomorphism. Therefore, $\mathcal{Q}$ belongs to $\mathcal{T}_1$. This implies that the category $\mathcal{T}_1$ is triangulated.
%
%It is clear that $\mathcal{T}_1$ is closed under isomorphisms. Let $\cM \in \mathcal{T}_1$. Suppose there exists $\mathcal{P}, \, \mathcal{Q} \in \Der^b_{\coh}(\cO_1)$ such that $\cM \simeq \mathcal{P} \oplus \mathcal{Q}$. Then $\Phi_{\cL}(\mathcal{P} \oplus \mathcal{Q}) \stackrel{\sim}{\to} \mathcal{P} \oplus \mathcal{Q}$ and $\Phi_{\cL}$ commutes with finite direct sums. It follows that
%$\Phi_{\cL}(\mathcal{P}) \stackrel{\sim}{\to} \mathcal{P}$ and $\Phi_{\cL}(\mathcal{Q}) \stackrel{\sim}{\to} \mathcal{Q}$. Thus, $\mathcal{T}_1$ is stable under taking direct summands. It follows that $\mathcal{T}_1$ is a thick subcategory of $\Der_{\coh}^b(\cO_1)$.
%
%Let $\mathcal{G}$ be a compact generator of $\Der_{\qcoh}(\cO_X)$. By Theorem \ref{BVdBcoh}, $\mathcal{G}$ classicaly generates $\Der_{\coh}^b(\cO_1)$. Since $\Phi_{\cK}$ is a equivalence we have the isomorphism
%\begin{equation*}
%\Phi_{\cK} \circ \Phi_{\cK_R}(\iota_g(\mathcal{G})) \stackrel{\sim}{\to}\iota_g(\mathcal{G}).
%\end{equation*}
%Applying the functor $\gr_\hbar$, we get that $\gr_\hbar \iota_g(\mathcal{G})$ belongs to $\mathcal{T}_1$ and by Lemma \ref{compcohdq}, $\gr_\hbar \iota_g(\mathcal{G})$ is a classical generator of 
%$\Der^b_{\coh}(\cO_1)$. Hence, $\mathcal{T}_1=\Der_{\coh}^b(\cO_1)$.
%Thus, the morphism \ref{isoeqi1} is an isomorphism. A similar argument shows that the morphism \ref{isoeqi2} is also an isomorphism which proves the claim.
%\end{proof}


\begin{thm}\label{finalmukai}
Let $X_1$ (resp. $X_2$) be a smooth complex projective variety endowed with a DQ-algebroid $\cA_1$ (resp. $\cA_2$). Let $\cK \in \Der^b_{\coh}(\cA_{12^a})$. The following conditions are equivalent
\begin{enumerate}[(i)]
\item The functor $\Phi_\cK: \Der^b_{\coh}(\cA_2) \to \Der^b_{\coh}(\cA_1)$ is fully faithful  (resp. an equivalence of triangulated categories).

\item The functor $\Phi_{\gr_\hbar \cK}:\Der^b_{\coh}(\cO_2) \to \Der^b_{\coh}(\cO_1)$ is fully faithful (resp. an equivalence of triangulated categories).
\end{enumerate}
\end{thm}

\begin{proof}
We recall the following fact. Let $F$ and $G$ be two functors and assume that $F$ is right adjoint to $G$. Then, there are two natural morphisms
\begin{align}
G \circ F \to \id \label{eq:1f}\\
\id \to F \circ G \label{eq:2f}.
\end{align}
%
The morphism (\ref{eq:1f}) (resp. (\ref{eq:2f})) is an isomorphism if and only if $F$ (resp. G) is fully faithfull. The morphism (\ref{eq:1f}) and (\ref{eq:2f}) are isomorphisms if and only if $F$ and $G$ are equivalences.

\begin{enumerate}
\item $(i) \Rightarrow (ii)$.
Proposition \ref{adjoint_mukai} is also true for $\cO$-modules since the proof works in the commutative case without any changes. Moreover, the functor $\gr_\hbar$ commutes with the composition of kernels. Hence, we have $\gr_\hbar (\cK_R) \simeq (\gr_\hbar \cK)_R$.
Therefore, the functor $\Phi_{\gr_\hbar \cK_R}$ is a right adjoint of the functor $\Phi_{\gr_\hbar \cK}$. Thus, there are morphisms of functors
\begin{align}
\Phi_{\gr_\hbar \cK} \circ \Phi_{\gr_\hbar \cK_R} \to \id, \label{isoeqi1}\\
\id \to \Phi_{\gr_\hbar \cK_R} \circ \Phi_{\gr_\hbar \cK}. \label{isoeqi2}
\end{align}
Set $\Phi_{\cL}=\Phi_{\gr_\hbar \cK_R} \circ \Phi_{\gr_\hbar \cK}$.
Let $\mathcal{T}_2$ be the full subcategory of $\Der^b_{\coh}(\cO_2)$ whose objects are the $\cM \in \Der^b_{\coh}(\cO_2)$ such that
\begin{equation*}
\cM \to \Phi_{\cL}(\cM)
\end{equation*}
is an isomorphism. It follows from Proposition \ref{thickiso} that $\mathcal{T}_2$ is a thick subcategory of $\Der_{\coh}^b(\cO_2)$.

Let $\mathcal{G}$ be a compact generator of $\Der_{\qcoh}(\cO_2)$. By Lemma \ref{compcohdq}, $\Der_{\coh}^b(\cO_2)=\langle \mathcal{G} \rangle$. Since $\Phi_{\cK}$ is a fully faithful we have the isomorphism
\begin{equation*}
\iota_g(\mathcal{G}) \stackrel{\sim}{\to} \Phi_{\cK_R} \circ \Phi_{\cK}(\iota_g(\mathcal{G})).
\end{equation*}
Applying the functor $\gr_\hbar$, we get that $\gr_\hbar \iota_g(\mathcal{G})$ belongs to $\mathcal{T}_2$ and by Lemma \ref{compcohdq}, $\gr_\hbar \iota_g(\mathcal{G})$ is a classical generator of 
$\Der^b_{\coh}(\cO_2)$. Hence, $\mathcal{T}_2=\Der_{\coh}^b(\cO_2)$.
Thus, the morphism (\ref{isoeqi2}) is an isomorphism of functors. A similar argument shows that if $\Phi_{\gr_\hbar \cK}$ is an equivalence the morphism (\ref{isoeqi1}) is also an isomorphism which proves the claim.


\item $(ii) \Rightarrow (i)$. 

Since $\Phi_\cK$ and $\Phi_{\cK_R}$ are adjoint functors we have natural morphisms of functors
\begin{align*}
\Phi_{\cK} \circ \Phi_{\cK_R} \to \id, \\
\id \to \Phi_{\cK_R} \circ \Phi_{\cK}.
\end{align*}

If $\cM \in \Der^b_{\coh}(\cA_2)$, then we have

\begin{equation} \label{buttraite}
\cM \to \Phi_{\cK_R} \circ \Phi_{\cK}(\cM).
\end{equation}

Applying the functor $\gr_\hbar$, we get

\begin{equation}\label{equicom}
\gr_\hbar \cM \to \Phi_{\gr_\hbar \cK_R} \circ \Phi_{\gr_\hbar \cK}(\gr_\hbar \cM).
\end{equation}

If $\Phi_{\gr_\hbar \cK}$ is fully faithful, then the morphism (\ref{equicom}) is an isomorphism. The objects $\Phi_{\cK_R} \circ \Phi_{\cK}(\cM)$ and $\cM$ are cohomologically complete since they belongs to $\Der^b_{\coh}(\cA_2)$.
Thus the morphism (\ref{buttraite}) is an isomorphism that is to say
\begin{equation*}
\id \stackrel{\sim}{\to} \Phi_{\cK_R} \circ \Phi_{\cK}.
\end{equation*}%
It follows that $\Phi_\cK$ is fully faithful.


Similarly, one shows that if $\Phi_{\gr_\hbar \cK}$ is an equivalence then in addition
\begin{equation*}
\Phi_{\cK} \circ \Phi_{\cK_R} \stackrel{\sim}{\to} \id.
\end{equation*}
It follows that $\Phi_{\cK}$ is an equivalence.
\end{enumerate}
%It is a direct consequence from Proposition \ref{stepmukaiun} and \ref{stepmukaideux}.
\end{proof}

\begin{Rem}
The implication $(ii) \Rightarrow (i)$ of Theorem \ref{finalmukai} and Proposition \ref{adjoint_mukai} still hold if one replaces smooth projective varieties by complex compact manifolds.
\end{Rem}